curate prediction of dispersion in both slow- and fast-mixing the theory Of dispersion in segsystems, thereby mented flow through open tubes.
SYMBOLS See list a t end of (16),which applies to both papers. LITERATURE CITED (1) L. J. Skeggs. Am. J. Clin. Pathol., 28, 311 (1957). (2) “Technicon Bibliography, 1967-75”. Technicon carp,, ~ ~~ , y , , 1975. (3) M, J, E, slay, in Chromatography, 1958jt,D, H, Des.,,, Ed, Academic Press, New York, 1958. p 36. (4) R . Thiers and K. Oglesby, Clin. Chem. ( Winston-Salem, N.C.), 10, 246 (1 964). (5) R. Thiers, R. Cole, and W. Kirsch, Clin. Chem. ( Winston-Salem, N.C.), 13, 451 (1967). (6) M. Evanson, G. Hicks, and R . Thiers. CM. Chem. ( Winston-Salem, N.C.). 16, 606 (1970). (7) R. Thiers, M. Jevn and R . Wildermann. Clln. Chem. ( Winston-Salem, N.C.), ’ 16, 832 (1970); (8)W. H. C. Walker, C. A. Pennock, and G. K. McGowan, Ciin, Chim. Acta, 27, 421 (1970).
(9) A. L. Chaney, in “Automation in Clinical Chemistry, 1967, Symposium I ” , N. B. Scova et ai., Ed., Mediad, Inc.. White Plains, N.Y., 1968, p 115. (10) J. Hrdina, 6th Colloq. Amino Acid Analysis, Technicon Corp.. Monograph No. 3, 1967. ( 1 1) G. Ertingshausen, H. J. Adler, and A. S. Reichler, J. Chromatogr., 42,355 (1969). (12) R . Thiers. A. Reed and K. Delander, Clin. Chem. ( Winston-Salem, N.C.), 17, 43 (1971). (13) R. Begg, Anal. Chem., 43, 854 (1971). (14) R . Begg. Anal. Chem., 44, 631 (1972). (15) W. H. C. Walker and K. R . Andrew, Clln. Chlm. Acta, 57, 181 (1974). (16) L. R . Snyder and H. J. Adler, Anal. Chem., 48, 1022 (1976). and Chem. Eng. Soc., 5, 258~(1956). , ~(17) A. Klinkenberg ~ ~ F. F. Sjenitzer, t ~ ~ (18) P. J. Concus, J. Phys. Chem., 74, 1818 (1970). (19) V. J. Levich, “Physicochemical Hydrodynamics”, Prentice Hall, Englewood Cliffs, N.J., 1962, p 681.
L; Fcifi;;ba,odt~~~ :i:g: ::?&{;,
~ ~ , ~ ; ; ~ (1974), ~ ~ , ~ (22) A. J. p. and L. M, Synge9 J.* 359 1358 (lg4’). (23) R. C. Weast, Ed., “Handbook of Chemistry and Physics”, 52nd ed.,Chemical Rubber Publishing Co., Cleveland, Ohio, 1971, p F-29. of Chemistry and Physics,,,31st &,, (24) c, D, Hodgeman, Rubber Publishing Co., Cleveland, Ohio, 1949, p 1768.
RECEIVED for review December 19,1975.Accepted February 18, 1976.
Dispersion in Segmented Flow through Glass Tubing in Continuous-Flow Analysis: The Nonideal Model L. R. Snyder” and H. J. Adler Technicon lnstruments Corp., Tarrytown, N.Y. 7059 7
The “ideal” model of sample dispersion in the flow of airsegmented liquid streams through open tubing is expanded here to include the effects of slow mixing within moving liquid segments. The resulting “nonideal” model allows predictlon of sample dispersion in segmented flow over a broad range of experimental conditions. Comparlson of experimental dispersion data with values calculated from the nonldeal model shows excellent agreement. The design of continuous-flow ( AutoAnalyzer) systems for minimum dispersion and sample interaction, and/or maximum analysis rates can now be done theoretically, rather than empirically as in the past.
In ( I ) , we described the development and experimental verification of a rigorous treatment of sample dispersion during segmented flow through open tubes: the so-called ideal model. The ideal model allows the approximate prediction of sample dispersion as a function of experimental conditions in continuous-flow (CF) or AutoAnalyzer analysis. This model is relatively accurate under conditions that allow near-complete mixing of sample within each liquid segment; however, it becomes less reliable when experimental conditions change so as to favor incomplete mixing. Here we expand the ideal model to correct for slow mixing, arriving finally a t the nonideal model. We also demonstrate its validity and ability to accurately predict dispersion over a range of conditions of interest in continuous-flow analysis.
EXPERIMENTAL E x p e r i m e n t a l conditions a n d procedures are described fully ( I 1. Unless otherwise noted, t h e standard conditions o f T a b l e I of (I) were used in each experiment. Precision a n d r e p r o d u c i b i l i t y are discussed in ( I ) .
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ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
THEORY A detailed, rigorous treatment of mixing within individual segments during segmented flow through open tubing has so far not been reported. Horvath e t al. ( 2 , 3 )have reported experimental data for intra-segment mixing during flow through enzyme-coated tubes, and correlated their results in terms of conventional engineering parameters such as the Nusselt and Reynolds numbers. However, their studies were directed toward radial mixing a t steady state, rather than axial dispersion. Here we will pursue an alternative approach, based on the theory of dispersion in chromatography. Bolus Flow. An understanding of mixing in segmented flow must begin with the characteristic bolus flow pattern that is observed within moving segments ( 2 , 4 , 5 ) .Figure 1 shows a schematic of this flow pattern, which is readily observable by adding dye at different points within the moving liquid segment. If dye is added a t the center A or leading edge B of the moving segment, the color of the dye is immediately dispersed into a characteristic figure-8 pattern, shown schematically by the crosshatched portion of Figure 1 (labeled ‘‘1”). After this figure-8 pattern is achieved, dye moves more slowly into adjacent regions or streamlines (“2”, “3”, etc.). Eventually the mixing of dye within the segment is complete, so that the dye is uniformly distributed. Thus, longitudinal mixing across the segment is rapid, while radial mixing or mass transfer is slow. A Hypothetical Analog of Mixing in Bolus Flow. The visual examination of dyed segments as above suggests the existence of discrete-essentially laminar-streamlines or currents ( 1 , 2 , . , . as in Figure 1).Mixing must then occur by movement of dye from one parallel current to its adjacent neighbor, by some combination of molecular diffusion plus convective mixing. This process appears conceptually similar
~
,
3
tention factor, r is the tube radius, u is the linear velocity of the liquid stream, D,, is the sample diffusion coefficient in that liquid, and L is tube length. We assume that Equation 3 is valid for bolus flow as well, if a) the diffusion coefficient D , is replaced by an effectiue diffusion coefficient D,’ that includes the additional mass transfer between streamlines as a result of convective mixing, and b) only one of the two symmetrical lobes of bolus flow (Figure 1)is compared with slug flow. The variance of Equation 3 expressed in length units can be related to the variance ur2through the relationship (34
ur,x = g r L s
Figure 1. Diagrammatic representation of the flow patterns within a moving liquid segment (Bolus flow)
to the hypothetical case of slug flow; Le., laminar flow through an open tube with the velocity of all streamlines constant (in actual laminar flow, the velocity of the liquid varies from zero a t the tube wall to a maximum velocity a t the tube center). In slug flow, mixing occurs by simple diffusion across adjacent streamlines (see discussion of 6). Although the symmetrical streamlines (e.g. No. 1 in Figure 1)are connected a t the ends of t,he segment in bolus flow, and are separated in slug flow, they are still part of the same concentric shell in slug flow. Therefore this difference in bolus vs. slug flow is of little real importance, and we will assume that radial mixing in each flow process is essentially similar. Finally, since longitudinal mixing is rapid in bolus flow, this effect does not alter the dispersion predicted by the ideal model, and can be ignored in further discussion here. A Model for Dispersion in Bolus Flow. Total sample dispersion can be defined in terms of the dimensionless variance uz of the resulting sample-concentration distribution (see 1 ). This total variance is the result of variances a ) predicted by the ideal model ui2 and b) as a result of slow mixing ur2. With the assumption that ui and rrare independent, we can write u2
=
ui2
= (I f
+
(1)
H* - 1=
ar2//q
(la)
= ur,x2/L= (1 - R ) 2 r 2 ~ / 4 D ,
The effective radius (or average diffusion distance) of the hemi-circle that comprises the cross-section of a single bolus lobe is related to tube internal diameter dt, and must have a value between d t / 2 and dJ4; or r = dt/3
(3c)
The combination of Equations 3, 3a-c then yields or2 = LV~’dt2~/36DmLs2V’
(4)
The fundamental parameter q = V f / V , was defined and discussed in ( I ) , where V , is the volume of a liquid segment. .41so. V , = VL,/L. These equations for q and V , plus Equation 4 give cr,’/q = Vrdt2u/36D,L V ,
(4a)
V f is related t o film thickness d fas
V f= TdtdfL
(4b)
d f= 0.5~d,(uq/y)”“
(4c)
Combining Equations 4a-c, with substitution of D,’ gives
for D,,
ur2/q = ( ~ ~ / 7 2 ) d t ~ ~ ~ ’ ~ =~ H* ~ ’ -~ 1/ ~ ~( 5 ’) ~ V ~ D ~ ’
(2)
The dispersion ur which arises from slow mixing in slug flow has been treated by Giddings (Equations 4.5-18 of 6) for the analogous case of chromatography in coated tubes. The tube coating or film is assumed to be quite thin, so that movement of sample between the film and adjacent streamline (e.g. No. 1of Figure 1)is rapid. In this sense, the transfer of sample (or dye) from the film into the adjacent streamline is exactly equivalent to the corresponding transfer of sample from film to streamline in bolus flow. For slug flow Giddings (6) gives
H
(3b)
q is liquid viscosity, and 7 liquid surface tension.
In ( I ) we defined the parameter H * ,
H * = u2/q Combining Equations 1 and l a gives
(1 - H ) = VfiV
where dfis given ( I ) as
Ur’i
ur2
L , is the length of the liquid segment: The quantity R in chromatography is the fraction of total sample (within the column or tube) that is contained in the moving liquid phase or segment. For the model of Giddings, where the coating exists under both air and liquid segments, its volume will be Vf. The equivalent volume of moving liquid within the tube will equal the tube internal volume, less Vf: V - Vf. Since the liquid in both coating (or film) and moving liquid are the same in segmented flow, the fraction R of sample in the moving liquid is simply ( V - V f ) / V ,and therefore
(3) Here H is the so-called plate-height contribution to dispersion from slow mixing (slow mass transfer) within the moving liquid, and ur,y2is the variance expressed in length units (along the tube), rather than in dimensionless units ur2. R is a re-
With the previous expression for q from ( I ) , q = 0.5TLd,2(u~/y)2’”Vs
(6)
we have in Equations l a , 5, and 6, a complete description of sample dispersion, with slow intra-segment mixing accounted for.
EXPERIMENTAL VERIFICATION OF THE NONIDEAL MODEL OF DISPERSION IN SEGMENTED FLOW THROUGH OPEN TUBES Values of H* and u, measured as in ( I ) , are given in Table I. These data correspond to the same experiments described in Table I11 of ( I ) ; each set of data in Table I had only one experimental parameter varied, which allows the testing of Equation 6 for that parameter. If Equation 5 is valid, derived values of the effective diffusion coefficient D,’ should be ANALYTICAL CHEMISTRY, VOL. 48, NO. 7 , JUNE
1976
1023
Table I. Variation of H* and Dispersion u with Experimental Conditions; Conditions as in Table I of ( I ) unless Otherwise Noted H* Variable T u b e length L
U
105 x D,' (calcd)
Value of variable
Exptl
Calcd"
Exptl
Calcd"
152 cm 305 610 1220
1.33 1.43 1.43 1.33
1.15 1.45 1.45 1.45
3.9 3.0 3.0 3.9 (3.4)e
1.14 1.55 2.32 3.08
1.10 1.55 2.19 3.10
b
Liquid velocity u, d , = 0.222 cm, V , = 0.067 ml
0.22 cm/s 0.37 0.74 I .10 1.47 2.21
1.20 1.13 1.51 2.33 2.32 3.37
1.06 1.14 1.44 1.87 2.42 3.78
0.8 3.1 2.5 1.8 3.0 3.3 (2.8)e
1.65 1.87 2.76 3.92 4.36 5.96
1.52 1.87 2.65 3.46 4.34 6.20
Liquid velocity u, L = 152 cm, ( d t = 0.159 cm )
0.83 cm/s 1.66 2.48 3.31
1.24 1.87 3.06 3.49
1,29 1.92 2.79 3.89
3.4 3.0 2.5 3.3 (3.0)e
1.23 1.94 2.88 2.85
0.94 1.45 2.00 2.59
Liquid velocity u, id, = 0.159 cm)
0.50 cm/s 0.83 1.66 2.48 3.31 4.97
1.16 1.48 2.02 2.60 3.73 6.13
1.12 1.29 1.92 2.79 3.89 6.65
2.22 2.82 4.11 5.33 7.10 10.4
2.10 2.67 4.10 5.66 7.35 11.0
Liquid velocity u , dt = 0.089 cm, L = 411 cm
1.26 cm/s 2.12 4.24 6.36 8.48 10.6 12.7
1.19 1.37 1.95 2.53 2.46 2.74 2.90
1.06 1.14 1.45 1.89 2.42 3.07 3.81
2.2 1.7 2.6 3.2 3.0 3.2 (3.0)e 0.9 1.0 1.3 1.6 2.7 3.2 4.0 (2.7)e
1.00 1.33 2.09 2.47 2.54 3.11 3.37
0.91 1.11 1.58 2.07 2.58 3.13 3.70
Liquid velocity LL, d t = 0.089 cm, L = 411 cm, V , = 0.10 ml
1.26 cm/s 2.12 4.24 6.36 8.48 12.72
1.13 1.15 1.09 1.51 1.39 1.76
1.02 1.05 1.15 1.29 1.46 1.91
0.4 0.8 4.5 1.5 3.3 3.3 (2.7Ie
0.65 0.71 0.75 1.15 1.23 1.52
0.50 0.61 0.81 0.98 1.15 1.51
Surface tension 7
32 dyne/cm 49.
1.41 1.27 1.53 1.18 1.30
1.45 1.34 1.31 1.27 1.26
3.2 3.6 1.7 4.4 2.5 (3.1)e
3.08 2.80 2.92 2.22 2.60
3.10 2.58d 2.44d 2.27d 2.1gd
3.21 2.09 1.57 1.25 1.05
2.38 2.06 1.55 1.37 1.18
1.8 2.8 2.8 4.3
7.39 5.08, 3.78 2.65 1.70
6.97 5.67 3.56 2.73 1.79
3.02 2.80 2.84 2.81
3.10 3.10 3.10 3.10
56 67 73 Liquid segment volume V ,
0.0107 ml 0.014 0.0267 0.040
0.080
(10) (2.6)e
Air segment volume Va
0.005 ml 0.008 0.014 0.020
1.37 1.30 1.28 1.35
1.45 1.45 1.45 1.45
3.5 4.3 4.6 3.7 (4.0)e
3.08 0.89 CP 1.51 2.5 3.76 1.15 2.0 1.76 4.59 1.4 1.33 2.24 5.45 1.54 1.0 2.80 7.65 2.16 0.7 4.40 a Equation 5, D,' = 2.9 X crn'isec. From experimental H* values via Equation. 5. As described in text, using Equations l a , 5, a n d 6. Assumes D = 0.50; D = 0.59 gives better results. e Weighted-average value.
Liquid viscosity
n
1024
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
I 1000
d,
=
35,,
1 105 D’,
1.0
.1 .01 I
1
,
,
1
, , , , , I
1.0
10
10
Figure 4. Dependence of Dm’on D, (different samples, see text)
U
Figure 2. Dependence of H” on
.1 105 D,
u. Data of Table I
V dt = 0.222, W di = 0.159. 0 Cq = 0.089
Table 11. D,’ Values as a Function of Sample Type Sample KMn04 Evans Blue Blue Dextrand
Mol wt 158 960 2 X lo6
Dm‘ 8.3 X 10-5 2.9 x 10-5 2.1 X 10-5
Drn 1.9 X 0.35 x 10-5 0.015 X 1 0 4c
b
Assumed equal to D, for similar compound KC104 (9). Wilke-Chang Equation ( 7 ) . Assumed equal to 11, for protein of similar mol wt (10). Nonionic dye, Pharmacia Corp,, Piscataway, N.J.
105 D ~ ’
1
2
3
4
5
(c Poise)
Figure 3. Dependence of Dm’on liquid viscosity. Data of Table I
constant as different parameters are varied. We have tabulated weighted-average values of D,’ for each data set in Table I (weighted by the factor H* - 1,see Equation 5 ) . We see in Table I that values of D,’ remain reasonably constant as different parameters are varied: tube length, liquid velocity, surface tension, liquid segment volume, and air segment volume. Weighted-average values of D,’ are also constant as tube diameter and liquid segment volume are varied (D,’ = 2.7, 3.0, and 2.8 X for dt = 0.089,0.159, and 0.22, respectively). Equation 5 predicts that (H* - I)V,/dt4 will vary with u5I3,and this relationship is confirmed in the log-log plot of Figure 2, where the theoretical straight line of slope 5 / 3 describes the data well. An overall weighted-average value of D,’ equal to 2.9 X 10-5 is calculated from Table I (excluding data for varying viscosity, see below). This compares with an estimated value of D, (Wilke-Chang equation, see 7) of 0.35 X 10-j.Thus, because of convective mixing in bolus flow, D,‘ for Evans Blue dye as sample is significantly greater than D,. Using this best value of D,’ (2.9 X loW5), values of H* for the experiments of Table I can be calculated. As seen in Table I, there is good general agreement between experimental and calculated H * values; an overall coefficient of variation equal to f15% is found. Effect of Liquid Viscosity. As 7 is varied in Table I from 0.89-2.16 cP, D,’ decreases from 2.5 to 0.7 X This is
expected, since it is known that D, varies with liquid viscosity. For example, the Stokes-Einstein equation (8) predicts that D , will vary as 1/7. The D,’ values of Table I are plotted (log-log)vs. 7 in Figure 3, with inclusion of additional dat,afor some organic so!vents. The best fit of these data by a linear curve suggests a slope of -1.7, or a dependence of D,’ on 7-l 7 . This strong dependence of d,’ on 7 is believed partially due to viscosity-inhibited momentum transfer within the moving liquid segment. We can modify Equation 5 to reflect this dependence of H * on 7 : D,‘ = Dw,~5‘(~/0.0089)-1~7
(7)
Here Dw,25’refers to the value of D,’ a t 25 “C wit,h water as liquid. Effect of Sample Type. As the molecular size of the sample compound is decreased, with corresponding increase in D,, D,‘ should approach D,, because diffusion is then favored with respect to convective mass transfer. Likewise, as D, becomes small, D,’ should approach a constant, value. Limited data for samples other than Evans Blue were obtained in the present study, values of D,’ were extracted as above, and these were compared to the average (best) value of D,‘ for Evans Blue. The resulting D,’ values are shown in Table 11. These data are also plotted in Figure 4, where the predicted dependence of D,’ on D, is observed. Figure 4 can he used to estimate values of D,‘ for other sample/liquid combinations. A value of D,‘ for st,raight rather than coiled tubing is included in Figure 4 (open square). Interestingly, this latter value of D,’ (0.48 X lo-”) is close to the value of D , (0.35 X lo-”), suggesting that diffusive mass transport is much more important than convective mass transfer in effecting mixing in flow through straight tubes. This is not unexpected (see ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1025
1.0
---
here. Nevertheless, the present treatment should allow reasonable accuracy in most cases. As a result, we are now in a position to examine the practical consequences of dispersion in different segmented-flow applications. For more details and one example of such an analysis, see ( 1 5 ) .
I
ol
APPENDIX I Correction to the Present Model for Small Values of u. The present model assumes that under suitable circumstances (e.g., very small values of u and v ) , the dispersion u can approach zero. However, even when the film thickness d fis negligible, and mixing within the individual segments is rapid (&’ large), the sample will be distributed across the initial segment. This initial dispersion uo is given (e.g., 16) as
0.6
uo = I
0
2
4
6
8
10
The total dispersion, taking
k-
ut2
Figure 5. Sample distribution curve for the nonideal case
Conditions:tube length = 610 cm, other conditions as in Table I of ( 1). . , . theoretical curve, q = 4.1. - - - theoretical curve, q = 6.1. - curve for q = 6.1, displaced to left by 2.0 tinits. 0 experimental data.
discussion of 2), and suggests that straight tubes should be avoided, if minimum dispersion is desired. Effect of Other Variables. H* was independent of tube coil diameter d,, for 4 < d , d 35 cm; see Table Vof (I).Temperature was not varied during the present study, but it is assumed that the variation of H* with temperature can be predicted from corresponding variations in 7 and y . Pump pulsations has a negligible effect on H*,as does sample concentration (see 1 ) .
SHAPE OF THE DISPERSION CURVE The present (nonideal) model provides a good description of the overall sample dispersion curve. This is illustrated in Figure 5 for a representative run. The solid points are experimental data, and the solid curve through these points is calculated from Equation 1 of ( I ) . using calculated values of y (4.1) and u* (6.1) from the present nonideal model. The lefthand dashed curve in Figure 5 is that predicted by the ideal model, which assumes u2 = q = 4.1. The right-hand dashed curve is for u2 = y = 6.1. It should be noted that earlier tests of Equation 1of (I), e.g., (11-14) did not recognize the importance of identifying the segment for which k = 0 in the final dispersed sample distribution. In effect, such tests allowed the arbitrary displacement of the calculated curve to the right or left (as in Figure 5 ) and, therefore, did not discriminate between the applicability of the ideal or nonideal models. I t is probable that most such previous data actually show significant nonideality.
CONCLUSIONS The present two papers provide an overall model for sample dispersion in segmented flow through open tubes. This model has been shown to be consistent with experimental data from studies covering a wide range in the relevant variables, suggesting that it is now possible to estimate dispersion in segmented flow through open tubes with reasonable accuracy. This is illustrated in Table I, where calculated values of u agree with experimental values within an overall relative standard deviation of 110%. The accuracy of the present approach in predictions for other segmented-flow systems may be limited in practice by a) tube-to-tube differences as discussed in ( I ) and b) possible inexactness of the present model as various experimental parameters are varied over much wider limits than studied 1026
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE
1976
=
u2
dim uo into
account is then given as
+ uo2 = u2 + (1/12)
The observed dispersion ut should then be slightly larger than that calculated as described in this paper, especially for small values of u. The increase in u due to this effect is calculated equal to 15%for u = 0.5,4%for u = 1.0, and 1%for u = 2.0. The effect is negligible for large values of o. Calculated values of u in Table I are not corrected for this effect.
SYMBOLS USED IN PRESENT TWO PAPERS (I, b, c, d , e , f = coefficients of Equation 5a of ( I ) C = sample concentration in an initial segment (before dispersion); Equation 1 of ( 1 )
ck
= sample concentration in the kth segment after dispersion; Equation 1 of ( I ) CF = abbreviation for continuous-flou d , = coil diameter of tube (cm) d f = thickness of liquid film adjacent to air-bubbles (cm) d , = internal diameter of tube (cm) D’ = constant defined by Equation ci of ( I ) D , = diffusion coefficient of sample in flowing liquid (cm*/sec) D,’ = effective diffusion coefficient in segmented flow; cf. discussion preceding Equation 3a (cm*/sec) H * = coefficient which measures departure of dispersion in segmented-flow from ideality; equal to u2/q-see Equation 6 of ( I ) k = segment number; see Figure 3 of ( I ) I, = tube length (cm) I,, = length of a liquid segment, assuming cylindrical shape (cm) 9 = dimensionless dispersion parameter, defined by Equation 2 of ( I ): equal to the displacement of the sample distribution from the initial sample segment (s);see Figure 4 of ( I ) ; related to experimental conditions by Equation 6 R = a retention factor defined by Equation 3b r = diameter of equivalent tube in slug flow (cm); see Equation 3c u = linear velocity of moving segments (cm/sec) V = tube internal volume (ml),equal to 7rdt2L/4 V , = volume of an air-bubble (ml) Vf = volume of liquid coating the tube interior, assuming no liquid segments within the tube (ml); equal to r d d d , L V , = volume of liquid segment (ml), equal to L,xdt2/4 y = liquid surface tension (dyne/cm) 7 = liquid viscosity (Poise, unless otherwise stated) u = measure of spreading or dispersion of sample after segmented flow through given tube, equal to square root of variance; dimensionless and measured in number of liquid segments over which dispersion has occurred; see Figures 4 and 5 of ( I ) u, = value of u for ideal dispersion only; see Equation 1 ur = value of u for radial dispersion (nonideal model) only; see Equation 1 = value of uI measured in lengt,h units (cm); see Equation 3
ACKNOWLEDGMENT We thank B. Oberhardt and R. Stoy of Technicon, C. Horvath of Yale University, D. Saunders of the Union Oil Co., J. J . Kirkland of DuPont, and B. Karger of Northeastern University for helpful comments on the present two papers.
LITERATURE CITED ( 1 ) L. R . Snyder and H. J. Adler, Anal. Chem., 48, 1017 (1976). (2) C. Horvath. B. A. Solomon, and J.-M. Engasser, lnd. Eng. Chem., Fundam., 12, 43 (1973). (3) J.-M. Engasser and C. Horvath, lnd. Eng. Chem., Fundam., 14, 107 (1975). (4) H. Goldsmith and S.Mason, in "Rheology. Theory and Applications", F. Eirich, Ed., Vol. 4, Academic Press, New York, 1967, p 242. (5) G. I. Taylor, NuidMech., 10, 11 (1961). (6) J. C. Giddings, "Dynamics of Chromatography", Dekker, New York, 1965, p 154. (7) B. L. Karger, L. R. Snyder, and C. Horvath, "An Introduction to Separation Science", Wiiey-lnterscience, New York. 1973, p 78. (8) Ref. 7, p 76. (9) R. A. Robinson and R. N. Stokes, "Electrolyte Solutions", 2nd ed., But-
terworths, London, 1959. (10) K. Diem and C. Lentner, Eds., "Scientific Tables", 7th ed.,Ciba-Geigy Ltd., Basle. 1970, p 580. (11) G. Ertingshausen, H. J. Adler, and A. S.Reichler, J. Chromatogr., 42, 355 (1969). (12) R. Thiers, A. Reed, and K. Delander, Clin. Chem. ( Winston-Salem, N.C.), 17, 43 (1971). (13) R. Begg, Anal. Chem., 43, 854 (1971). (14) W. H. C. Walker and K. R. Andrew, Clin. Chim. Acta, 57, 181 (1974). (15) L. R. Snyder, J. Chromafog., in press. (16) J. C. Sternberg, Adv. Chromatogr., 2, 205 (1966)
RECEIVED for review December 19,1975. Accepted February 18, 1976.
Relationship between Mutual Information and Classification G. L. Ritter, S. R. Lowry, H. B. Woodruff, and T. L. Isenhour' Department of Chemistry, University of North Carolina, Chapel Hill, N.C. 27514
Mutual information Is an information theory term, which describes the amount of information to distinguish among members of different classes. This concept is demondated for describing the classification ability of a maximum likelihood classifier. in an artificlaily generated problem, the recognition ability and the square root of mutual information are seen to be linearly related. The linear correlation drops to 0.83 for a series of infrared questions where the statistical independence assumption is not strictly valid.
In most chemical problems, the goal is to find some transformation of measured experimental properties that reveals another more obscure chemical or physical property. Recently there has been interest in applying the principles of information theory to such chemical systems (cf. 1-3). Modern information theory is largely a result of work done by Claude Shannon in the 1940's ( 4 ) . Shannon was investigating communication systems where a message is transferred from a transmitter to a receiver. In this work, he assumed that the source of the message was stochastic, and he wished t o exactly or approximately reproduce the message a t the receiver. The solution to the problem required a definition of an uncertainty or "entropy" in the message a t the receiver. In chemical systems, the format of the information theory problem is usually similar. First, an experimenter measures an event, X ,which may take on any of several states, x i . The outcome of must fall into one of the states; call it Xk. The fraction of the time that X k occurs denotes a probability for the state, P k . Given this stochastic form for X ,information theory defines information or entropy as a measure of the uncertainty in predicting the state of X.Shannon suggested the following as a measure of the uncertainty for a system of n possible states.
x
Some other event, Y may occur in conjunction with X.A complete description of this system requires not only a probability scheme for the states of X ,but also a probability scheme for Y and a set of conditional probabilities. The conditional probability, denoted Pklj, indicates the probability of state x k arising when Y is in steady y j . From the complete
set of probabilities and conditional probabilities, the mutual information may be found. The mutual information is the amount of information in the event X t o determine the state of Y ( 5 ) .These definitions suggest the application of information theory to chemical classification problems. In the classification problem, we wish to distinguish or discriminate an obscure chemical or physical property by use of a mathematical transformation of the measured experimental quantities. This property classifier is normally designed to minimize the number of incorrect categorizations in a collection of known data. If the classifier is to do well, then there must be information in the experimental data to determine the unknown property. T o test the relationship between classification or discrimination and mutual information, we wish to choose a classifier that is stochastic and attempts to minimize the number of errors. This classifier is a maximum likelihood estimator. The spectroscopic data that we use to test the relationship are a series of substructure questions using infrared spectra. The spectra are coded in an intensity eliminated or peak/no peak form. From the experimentally measured binary infrared spectra, we attempt to estimate the amount of information to determine the various chemical substructures. In the classification decision, the category or substructure that is most likely to give the infrared spectrum of the unknown is selected. If the mutual information content is high, then the recognition ability should also be high.
DATA The data for this study have been taken from the American Society for Testing and Materials file of 91 875 binary infrared spectra. Thirteen mutually exclusive classes were chosen as the criterion for selecting spectra (Table I) (6).Of the spectra belonging exclusively to each of the classes, 200 were randomly chosen and placed into the data set. The resulting 2600 compounds each contain fewer than sixteen carbons and contain only hydrogen, oxygen, and nitrogen otherwise. Each compound is represented by a peak/no peak spectrum in the range 2.0 t o 15.9 pm. This range is divided into 139 equal-sized intervals, each describing whether a peak maximum occurs in the corresponding interval. For convenience, therefore, each spectrum is given as a 139-dimensional binary vector x such that ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
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